Ullman's Theorem in Two Dimensions

In the field of computer vision reconstructing an object using a video of its motion is known as the structure from motion problem. This Demonstration illustrates the reconstruction for three orthographic cameras and three points in the plane. The reconstruction is unique modulo the group of Euclidean transformations and reflections in the plane. The user can change the projection parameters. The inversion computes the camera and point configurations from these projections.

The first six controls change the picture data that the three cameras have taken from the two unknown points. The unit vectors , , are the the camera lines and A, B are the points. We know the six dot products of A and B with , , . From that, the structure from motion inversion determines the points A, B and the vectors , , . The last two sliders control the size of the graphic.